Vector space Definition and 540 Threads

Homework Statement Let V= set of 2x2 matrices with the normal addition, but where multiplication is defined as: β#A=β(A^T) where A^T is the transpose of A. Homework Equations The axiom about 1#A=A The Attempt at a Solution I think that because you can show that not ALL matrices...

Dear all, I am not very experienced in this field, so, I have a rather simple question :smile: -Consider a linear vector space V of dimension 4. -Prescribe that, if two vectors in V differ by a nonvanishing constant, they belong to the same equivalence class. -Put together all these...

According to my book, a vector space V is a set endowed with two properties: -closure under addition -closure under scalar multiplication and these two properties satisfy eight axioms, one of which is: "for all f in V there exists -f in V such that f+(-f)=0" But then isn't this axiom...

Hi. I attached the problem and my work. I'm not sure if I did part a) right. In the past problems I've done, they usually provide you with 3 vectors that are linearly independent, thus giving you unique values for C1, C2, C3. The matrix for this one forms: 1 1 1 0 1 3 0 0 0 Which is...

Homework Statement The set K of 2 × 2 real matrices of the form [a b, -b a] form a field with the usual operations. It should be clear to you that M22(R) is a vector space over K. What is the dimension of M22(R) over K? Justify your answer by displaying a basis and proving that the set...

Homework Statement The problems states "All polynomials of the form p(t)= at^2, where a is in R." I'm supposed to see if it is a subspace of Pn. I've already done that but the book's answer is that it spans Pn by Theorem 1, because the set is span{t^2} Homework Equations Theorem states "1 If...

So the title says everything. Let's assume R is a set equipped with vector addition the same way we add real numbers and has a scalar multiplication that the scalars come from the field Q. I believe the dimension of this vector space is infinite, and the reason is we have transcendental numbers...

Homework Statement So basically, I'm studying the proof for this: "In a finite dimensional vector space, the length of every linearly independent list of vectors is less than or equal to the length of every spanning list of vectors." What the book (Axler's Linear Algebra Done Right 2e) does...

Homework Statement Let V = {0,1} with addition defined modulo 2 (i.e. the remainder upon division by 2), and scalar multiplication given by ku = u^k for all k in the real numbers and u in V. Is the set V a vector space? Homework Equations The 10 axioms! The Attempt at a Solution I...

Every resource I've looked at just lists the axioms but doesn't tell how or why they were arrived at. To what extent are they arbitrary?

How can I prove that the set of all planar vector fields forms a vector space? Thanks for any input!

Homework Statement Is it correct to say that the dimension of a given vector space is equal to the number of vectors of the canonic solution? For example: Vector space |R3 Canonic solution = {[1 0 0],[0 1 0],[0 0 1]} Therefore its dimension is 3. Homework Equations The Attempt...

Homework Statement Let V be the set of all ordered pairs of real numbers, with addition being defined as: (x_1 , x_2 ) + (y_1 , y_2 ) = (x_1 + y_1 , x_2 + y_2 ) and scalar multiplication defined as: \alpha \circ (x_1 , x_2 ) = (\alpha x_1 , x_2) Is V a vector space with these...

Homework Statement Determine if the following set is a vector space under the the given operations. The set V of all pairs of real numbers of the form (1,x) with the operations: (1, y) + (1, y') = (1, y + y') k(1, y) = (1, ky) Homework Equations The Attempt at a Solution...

Homework Statement Determine if the following set is a vector space under the given operations. List all the axioms that fail to hold. The set of all pairs of real numbers of the form (x,y), where x >= 0, with the standard operations on R^2 Homework Equations The Attempt at a...

There's a geometric interpretation of the determinant of an operator in a real vector space that I've always found intuitive. Suppose we have a n-dimensional real-valued vector space. We can plot n vectors in an n-dimensional Cartesian coordinate system, and in general we'll have an...

Homework Statement Greetings, I'm trying to solve these problems given the vectors u=(3,-2,1) and v=(2,-3,1) 1. Find an orthogonal base for the space H generated by {u,v} 2. Find the orthogonal projection of w=(3,0,1) on H Homework EquationsThe Attempt at a Solution Im not sure how...

Homework Statement Show that the system S is not a vector space by showing one of the axioms is not satisfied with the usual rules for addition and multiplication by a scalar in ℝ3 S={x(ℝ3):2x1+3x32-4x23=0} Homework Equations The Attempt at a Solution The subscripts for the...

Homework Statement This is a problem from chapter 1.3 of Linear Algebra by F/I/S. Let W_{1} and W_{2} be subspaces of a vector space V. Prove that W_{1} \cup W_{2} is a subspace of V iff W_{1}\subseteqW_{2} or W_{2} \subseteq W_{1}. Homework Equations See attempt at solution. The...

I was working on a problem in field extensions (for a 3rd-year ring theory class), and came to a point where I essentially had the following situation... F is a field isomorphic to G , and G' is for all intents and purposes, some set. We can then consider the vector spaces G'_F and...

Homework Statement Let V = {RXR} with addition defined as (a1,a2) + (b1,b2) = (a1 + b1, a2b2) Show the vector space condition, for each x in V there exists a y in V such that x + y = 0, fails for the V defined above. Homework Equations The Attempt at a Solution a + d = (a1,a2)...

Hi everyone, I am having difficulty solving the following question: Prove that the set V=R^2 with addition defined by (x,y)+(x'+y')=(x+x'+1, y+y') and scaler multiplication by k(x,y)=(kx+k-1, ky) is a vector space, find-(x+y) and the zero vector in this vector space. the following was...

There is a part in the proof of sard's theorem where we restrict our discussion to a point x such that Df(x)=0, and then declare that f ' (x) is a proper (n-1) subspace (f is n-dim). What I don't understand is, the argument then goes by considering any two points in a sub-rectangle around this...

If I am given a coefficient matrix of m rows and n columns (an m x n matrix), then is the vector space of that matrix Rm or Rn? I get really confused sometimes. Sometimes, the superscript seems like the number of rows, and sometimes the number of variables. It also doesn't help that my class...

I am interested in knowing that in QM what vector/tensor operators are? In fact how do they differ from scalar operators?

Homework Statement Let V be a vector space and v, w two elements of V. If v+w=O, show that w=-v Homework Equations The Attempt at a Solution This is my attempt: 1. v+w=O 2. v+(-v)=O 3. v+w=v+(-v). Subtracting v from both sides yields: 4. w=-v The solution...

There is a difference between the multiplication op in the Field of Reals and the Scaling op in the Set of Vectors. In the Field, the op is a closed mapping between two members within the same Set, whereas in the Set of Vectors, the op is a mapping between a member of the Set of Vectors and a...

Homework Statement V1 is defined as the span of the vectors b1,b2,b3,b4 Prove that V1 is not a vector space. Homework Equations A set of axioms : If u and v are tow vectors in the span of b1,b2,b3,b4 then: 1. u + v belongs to V1 2. ku also belongs to V1 The Attempt at a...

Hi: I am a newbie to linear algebra; I have a problem about vector space and subspaces. How to distinguish these two subject. what I know from books is subspace is going through zero, but I still can not figure out what is the difference between vector space and subspaces, thanks.

I'm having a hard time visualising them. How do you plot vector fields How do you plot spanning sets? How do you tell if something spans a plane, 3d, a line or more dimensions.

Let w=span(w1, w2, ...,wk) where wi are vectors in R^n. Let dot product be inner product for R^n here. Prove that if v*wi=0 for all i-1,2,...,k then v is an element of w^upside down T (w orthogonal).

question in attachment. please help!

Sometimes the way authors write a book makes you wonder if they are at their wits' end: 'A vector space with an inner product is an inner product space!' I am not sure if I have gone crazy but to me it is obvious that if you have a vector space wrapped up in a nice gift basket and sent off to...

Homework Statement Consider the collection of all polynomials (with complex coefficients) of degree < N in x. a) Does this set constitute a vector space? If so, suggest a convenient basis, and give the dimension of the space. If not, which of the defining properties does it lack? b)...

Homework Statement Consider the ordinary vectors in three dimensions (ax, ay, az) with complex components. a) Does the subset of all vectors with az = 0 constitute a vector space? If so, what is its dimension; if not; why not? b) What about the subset of all vectors whose z component is 1...

Homework Statement V is an arbitrary vector space and v,w,x are part of V such that v + x = w = x Use vector space axioms to prove v = w I've looked at the axioms for an hour and can not get any lead to start this question.

Homework Statement prove 0u=0 where u is in the vector space. Homework Equations the 10 various axioms for addition and scalar multiplication. The Attempt at a Solution pretty much just (u+-u)u=0 or (1-1)u=0 1u+-1u=0 and then i get stuck. i can prove that -1u=-u but that...

-set of all nonnegative real numbers -set of all upper triangular nxn matrices -set of all upper triangular square matrices from the book, the answers are: yes, no(need to be the same size to define addition), and yes for the set of all nonnegative real numbers, doesn't it fail closure...

How do I show that for any vectors u,v, and w in a vector space V, the set of the vectors {u-v, v-w, w-u} is a linearly dependent set? do it in general!

Hi all I hope you guys can help me. I am soo confused with this question: I would really liked a complete answer to this, I have an upcoming exam and I know these two will be on the exam. 1. Let V be the set of all diagonal 2x2 matrices i.e. V = {[a 0; 0 b] | a, b are real numbers} with...

Homework Statement Consider the vector space l_infty(R) of all bounded sequences. Decide whether or not the following norms are defined on l_infty(R) . If they are, verify by axioms. If not, provide counter example. Homework Equations x in l_infty(R); x=(x_n), (i) || ||_# defined by...

Homework Statement 1) Consider the 3 norms in vector space R^3, ##\| \|_i## where i=1,2 and infinity. Given x = (2, -5,3) and y = ( -3, 2,0). Calculate ##\|x\|_1, \|x+y\|_2, \|x-2y\|_\infty## 2)Prove Rigorously that ##\displaystyle \lim_{n \to \infty}=\frac{4n^2+1}{2n^2-1}=2##Homework...

Question: Show the set of all differentiable functions on (-infinity, +infinity) that satisfy f′ + 2f = 0 is a vector space. I started the problem by assuming that f and g are both differentiable functions that satisfy this vector space. Then I ran through the ten axioms of addition and...

Homework Statement Let g_1(t) = t - 1 and g_2(t)= t^2+t. Using the inner product on P_2 defined in example 10(b) with t_1=-1,t_2=0,t_3=1, find a basis for the orthogonal complement of Span(g_1, g_2). Homework Equations From example 10(b) \langle p, q \rangle = \sum_{i=1}^{k+1}...

Hi there. I'm a long time reader, first time poster. I'm an undergraduate in Math and Economics and I am having trouble in Linear Algebra. This is the first class I have had that focuses solely on proofs, so I am in new territory. Homework Statement note Although the question doesn't state...

Homework Statement Consider the vector space R^2, Decide whether or not the following are norms defined on R^2. If they are, verify all axioms of a norm, if not, demonstrate by counter example some axioms which fails. For x=(x_1,x_2) in R^2 Homework Equations (i) || ||_#: R^2 defined...

Homework Statement A vector space W over the real numbers is the set of all 2 x 2 Hermitian matrices. Show that the map T defined as: T(x,y,z,t) = [t+x y+iz] [y-iz t-x] from R4 to W is an isomorphism. Homework Equations The Attempt at a Solution I know that for the map...

The general linear group of a vector space GL(V) is the group who's set is the set of all linear maps from V to V that are invertible (automorphisms). My question is, why is this a group? Surely the zero operator that sends all vectors in V to the zero vector is not invertible? But isn't it...

Homework Statement Determine if this set is a vector space. Either show that the necessary properties are satisfied, or give an example showing that at least one of them is not. -The set P of all polynomials, with the usual definition of scalar multiplication, but with addition defined...

Homework Statement Let (X,||\cdot||) be a normed vector space and suppose that Y is a closed vector subspace of X. Show that the map ||x||_1=\inf_{y \in Y}||x-y|| defines a pseudonorm on X. Let (X/Y,||\cdot||_1) denote the normed vector space induced by ||\cdot||_1 and prove that the...